Hough Pulsar

Author:
Alicia Sintes, Badri Krishnan Routines for building and updating the space of partial Hough map derivatives and related functions needed for the construction of total Hough maps at different frequencies and possible residual spin down parameters.
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Files

file  HoughMap.h
 
Author:
Alicia M.Provides subroutines for initialization and construction of Hough-map derivatives and total Hough-maps.

file  Statistics.h
 Functions prototypes and structures for calculating statistical properties of Hough maps.
file  Velocity.h
 
Author:
Badri Krishnan Header file for velocity and position routines.

Detailed Description

Author:
Alicia Sintes, Badri Krishnan Routines for building and updating the space of partial Hough map derivatives and related functions needed for the construction of total Hough maps at different frequencies and possible residual spin down parameters.

Description
As we mention before, the issue is to build histograms, the Hough map (HM), in the parameter space: for each intrinsic frequency $ f_0 $, each residual spin-down parameter, and each refined sky location inside the patch. Notice, from the master equation, that the effect of the residual spin-down parameter is just a change in $ F_0$ , and, at any given time, $ F_0 $ can be considered constant. Also, the Hough map is a histogram, thus additive. It can be seen as the sum of several partial Hough maps constructed using just one periodogram.

Therefore, we can construct the HM for any $ f_0$ and spin-down value by adding together, at different times, partial Hough maps (PHM) corresponding to different $ F_0$ values (or equivalently, adding their derivatives and then integrating the result).

In practice this means that in order to obtain the HM for a given frequency and all possible residual spin-down parameters, we have to construct a CYLINDER of around the frequency $ f_0$ . All of the phmd coming from data demodulated with the same parameters. The coordinates of the phmd locate the position of the source in the sky, and by summing along different directions inside the cylinder we refine the spin-down value. To analyze another frequency, for all possible spin-down parameters, we just need to add a new line to the cylinder (and remove another one, in a circular buffer) and then proceed making all the possible sums again.

For the case of only 1 spin-down parameter we have to sum following straight lines whose slope is related to the grid in the residual spin-down parameter. We can distinguish (at most) as many lines as the number of the different periodograms used.

Author:
Sintes, A.M, Papa, M.A. and Krishnan, B. File Name: LUT.h
Id
LUT.h,v 1.11 2007/02/26 13:16:26 badri Exp

History: Created by Sintes May 11, 2001 Modified by Badri Krishnan Feb 2003

Description
Our goal is the construction of Hough maps. In order to produce them efficiently, the present implemetation makes use of { lut}s. Here we provide the necessary routines for their construction and use.\

In principle, the subroutines provided are valid for { any} Hough master equation of the form: $ \nu-F_0 =\vec\xi (t) \cdot (\hat n -\hat N )\, ,$ where $\nu$ is the measured frequency of the signal at time $t$, $F_0$ intrinsic frequency of the signal at that time, $\hat n$ location of the souce in the sky, $\hat N$ the center of the sky patch used in the demodulation procedure, and $\vec\xi (t)$ any vector.

The form of this vector $\vec\xi (t)$ depends on the demodulation procedure used in the previous step. In our case this corresponds to $\vec\xi (t) = \left( F_0+ \sum_k F_k \left[ \Delta T \right]^k\right) \frac{\vec v(t)}{c} + \left( \sum_k k F_k \left[ \Delta T \right]^{k-1}\right) \frac {\vec x(t)- \vec x(\hat t_0)}{c}\, ,$ and $F_0 \equiv f_0 + \sum_k \Delta f_k \left[ \Delta T \right]^k \, , $ where $\vec v(t)$ is the velocity of the detector, $\vec x(t)$ is the detector position, $ T_{\hat N}(t)$ is the time at the solar system barycenter (for a given sky location $\hat N$), $\Delta T \equiv T_{\hat N}(t)-T_{\hat N}(\hat t_0)$, $\Delta f_k = f_k-F_k$ the residual spin-down parameter, $F_k$ the spin-down parameter used in the demodulation, and $f_0$, $f_k$ the intrinsic frequency and spin-down parameters of the source at time $\hat t_0$.\

Looking at the generic Hough master equation, one realizes that for a fixed time, a given value of $F_0$, and a measured frequency $\nu$ (from a selected peak), the source could be located anywhere on a circle (whose center points in the same direction of $\vec\xi (t)$ and is characterized by $\phi$, the angle between $\hat n$ and $\vec\xi$). Since the Hough transform is performed on a set of spectra with discrete frequencies, a peak on the spectrum appearing at $\nu$ could correspond to any source with a demodulated frequency in a certain interval. As a consequence, the location of the sources compatible with $F_0$ and $\nu$ is not a circle but an annulus with a certain width.\

Our purpose is to map these annuli on a discrete space. An estimation of the average thickness of the annuli tells us that the vast majority of annuli will be very thin, and therefore our algorithm should not be optimized for drawing thick annuli but for thin ones. Also, the mapping implementation should be one with a uniform probability distribution in order to avoid discretization errors. In order to remove border effects, we use a biunivocal mapping, which requires that a pixel in a partial Hough map can belong only to one annulus, just touched by one peak of the spectrum. The criteria for the biunivocal mapping is that if and only if the center of the pixel is inside the annulus, then the pixel will be enhanced.\

In order to simplify (reduce the computational cost of) this task we construct look up tables ({ lut}) where the borders of these annuli are marked for any possible $\nu -F_0$ value. Since we work on a discrete space these { lut} are valid for many $F_0$ values.\

Uses
LALHOUGHComputeSizePar ()
LALHOUGHComputeNDSizePar () 
LALHOUGHFillPatchGrid ()
LALHOUGHParamPLUT ()
LALNDHOUGHParamPLUT ()
LALRotatePolarU()
LALInvRotatePolarU()
LALStereoProjectPolar()
LALStereoProjectCart()
LALStereoInvProjectPolar()
LALStereoInvProjectCart()
LALHOUGHConstructPLUT()
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